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SEIL KIM,HYUNSHIG JOO,HAESEONG CHO,SANGJOON SHIN 한국산업응용수학회 2017 Journal of the Korean Society for Industrial and A Vol.21 No.3
In this paper, development of the three-dimensional structural analysis is performed by applying FETI-local method. In the FETI-local method, the penalty term is added as a preconditioner. The OPT-DKT shell element is used in the present structural analysis. Newmark- method is employed to conduct the dynamic analysis. The three-dimensional FETI-local static structural analysis is conducted. The contour and the displacement of the results are compared following the different number of sub-domains. The computational time and memory usage are compared with respect to the number of CPUs used. The three-dimensional dynamic structural analysis is conducted while applying FETI-local method. The present results show appropriate scalability in terms of the computational time and memory usage. It is expected to improve the computational efficiency by combining the advantages of the original FETI method, i.e., FETI-mixed using the mixed local-global Lagrange multiplier.
Kim, H.H.,Lee, C.O.,Park, E.H. Pergamon Press ; Elsevier Science Ltd 2010 COMPUTERS & MATHEMATICS WITH APPLICATIONS - Vol.60 No.12
Selection of primal unknowns is important in convergence of FETI-DP (dual-primal finite element tearing and interconnecting) methods, which are known to be the most scalable dual iterative substructuring methods. A FETI-DP algorithm for the Stokes problem without primal pressure unknowns was developed and analyzed by Kim et al. (2010) [1]. Only the velocity unknowns at the subdomain vertices are selected to be the primal unknowns and convergence of the algorithm with a lumped preconditioner is determined by the condition number bound C(H/h)(1+log(H/h)), where H/h is the number of elements across subdomains. In this work, primal unknowns corresponding to the averages on edges are introduced and a better condition number bound C(H/h) is proved for such a selection of primal unknowns. Numerical results are included.
A dual iterative substructuring method with a penalty term in three dimensions
Lee, C.O.,Park, E.H. Pergamon Press ; Elsevier Science Ltd 2012 COMPUTERS & MATHEMATICS WITH APPLICATIONS - Vol.64 No.9
The FETI-DP method is one of the most advanced dual substructuring methods, which introduces Lagrange multipliers to enforce the pointwise matching condition on the interface. In our previous work for two dimensional problems, a dual iterative substructuring method was proposed, which is a variant of the FETI-DP method based on the way to deal with the continuity constraint on the interface. The proposed method imposes the continuity not only by the pointwise matching condition on the interface but also by using a penalty term which measures the jump across the interface. In this paper, a dual substructuring method with a penalty term is extended to three dimensional problems. A penalty term with a penalization parameter η is constructed by focusing on the geometric complexity of an interface in three dimensions caused by the coupling among adjacent subdomains. For a large η, it is shown that the condition number of the resultant dual problem is bounded by a constant independent of both subdomain size H and mesh size h. From the implementational viewpoint of the proposed method, the difference from the FETI-DP method is to solve subdomain problems which contain a penalty term with a penalization parameter η. To prevent a large penalization parameter from making subdomain problems ill-conditioned, special attention is paid to establish an optimal preconditioner with respect to a penalization parameter η. Finally, numerical results are presented.
A DUAL ITERATIVE SUBSTRUCTURING METHOD WITH A SMALL PENALTY PARAMETER
Lee, Chang-Ock,Park, Eun-Hee Korean Mathematical Society 2017 대한수학회지 Vol.54 No.2
A dual substructuring method with a penalty term was introduced in the previous works by the authors, which is a variant of the FETI-DP method. The proposed method imposes the continuity not only by using Lagrange multipliers but also by adding a penalty term which consists of a positive penalty parameter ${\eta}$ and a measure of the jump across the interface. Due to the penalty term, the proposed iterative method has a better convergence property than the standard FETI-DP method in the sense that the condition number of the resulting dual problem is bounded by a constant independent of the subdomain size and the mesh size. In this paper, a further study for a dual iterative substructuring method with a penalty term is discussed in terms of its convergence analysis. We provide an improved estimate of the condition number which shows the relationship between the condition number and ${\eta}$ as well as a close spectral connection of the proposed method with the FETI-DP method. As a result, a choice of a moderately small penalty parameter is guaranteed.
Kim, Hyea Hyun,Chung, Eric,Wang, Junxian Elsevier 2017 Journal of computational physics Vol.349 No.-
<P><B>Abstract</B></P> <P>BDDC and FETI-DP algorithms are developed for three-dimensional elliptic problems with adaptively enriched coarse components. It is known that these enriched components are necessary in the development of robust preconditioners. To form the adaptive coarse components, carefully designed generalized eigenvalue problems are introduced for each faces and edges, and the coarse components are formed by using eigenvectors with their corresponding eigenvalues larger than a given tolerance <SUB> λ T O L </SUB> . Upper bounds for condition numbers of the preconditioned systems are shown to be C <SUB> λ T O L </SUB> , with the constant <I>C</I> depending only on the maximum number of edges and faces per subdomain, and the maximum number of subdomains sharing an edge. Numerical results are presented to test the robustness of the proposed approach.</P>
Min-Ki Kim,Seung-Jo Kim 한국항공우주학회 2008 International Journal of Aeronautical and Space Sc Vol.9 No.2
High performance direct-iterative hybrid linear solver for large scale finite element problem is developed. Direct solution method is robust but difficult to parallelize, whereas iterative solution method is opposite for direct method. Therefore, combining two solution methods is desired to get both high performance parallel efficiency and numerical robustness for large scale structural analysis problems. Hybrid method mentioned in this paper is based on FETI-DP (Finite Element Tearing and Interconnecting-Dual Primal method) which has good parallel scalability and efficiency. It is suitable for fourth and second order finite element elliptic problems including structural analysis problems. We are using the hybrid concept of theses two solution method categories, combining the multifrontal solver into FETI-DP based iterative solver. Hybrid solver is implemented for our general structural analysis code, IPSAP.
Kim, Min-Ki,Kim, Seung-Jo The Korean Society for Aeronautical and Space Scie 2008 International Journal of Aeronautical and Space Sc Vol.9 No.2
High performance direct-iterative hybrid linear solver for large scale finite element problem is developed. Direct solution method is robust but difficult to parallelize, whereas iterative solution method is opposite for direct method. Therefore, combining two solution methods is desired to get both high performance parallel efficiency and numerical robustness for large scale structural analysis problems. Hybrid method mentioned in this paper is based on FETI-DP (Finite Element Tearing and Interconnecting-Dual Primal method) which has good parallel scalability and efficiency. It is suitable for fourth and second order finite element elliptic problems including structural analysis problems. We are using the hybrid concept of theses two solution method categories, combining the multifrontal solver into FETI-DP based iterative solver. Hybrid solver is implemented for our general structural analysis code, IPSAP.
YUNSHIG JOO,DUHYUN GONG,SEUNG-HOON KANG,TAEYOUNG CHUN,신상준 한국산업응용수학회 2020 Journal of the Korean Society for Industrial and A Vol.24 No.2
This paper describes the development of a parallel computational algorithm based on the finite element tearing and interconnecting (FETI) method that uses a local Lagrange multiplier. In this approach, structural computational domain is decomposed into non-overlapping sub-domains using local Lagrange multiplier. The local Lagrange multipliers are imposed at interconnecting nodes. 8-node solid element using extra shape function is adopted by using the representative volume element (RVE). The parallel computational algorithm is further established based on message passing interface (MPI). Finally, the present FETI-local approach is implemented on parallel hardware and shows improved performance.
Lee, Chang-Ock,Park, Eun-Hee,Park, Jongho Korean Mathematical Society 2021 대한수학회지 Vol.58 No.3
In this corrigendum, we offer a correction to [J. Korean Math. Soc. 54 (2017), No. 2, 461-477]. We construct a counterexample for the strengthened Cauchy-Schwarz inequality used in the original paper. In addition, we provide a new proof for Lemma 5 of the original paper, an estimate for the extremal eigenvalues of the standard unpreconditioned FETI-DP dual operator.
A NON-OVERLAPPING DOMAIN DECOMPOSITION METHOD FOR A DISCONTINUOUS GALERKIN METHOD: A NUMERICAL STUDY
Eun-Hee Park The Kangwon-Kyungki Mathematical Society 2023 한국수학논문집 Vol.31 No.4
In this paper, we propose an iterative method for a symmetric interior penalty Galerkin method for heterogeneous elliptic problems. The iterative method consists mainly of two parts based on a non-overlapping domain decomposition approach. One is an intermediate preconditioner constructed by understanding the properties of the discontinuous finite element functions and the other is a preconditioning related to the dual-primal finite element tearing and interconnecting (FETI-DP) methodology. Numerical results for the proposed method are presented, which demonstrate the performance of the iterative method in terms of various parameters associated with the elliptic model problem, the finite element discretization, and non-overlapping subdomain decomposition.